Research interests

Uses and meanings of mathematics in England, 1650-1750

When and how did the concept of 'applied mathematics' arise? The term came into use in the early nineteenth century, following a long process of change in the uses and perceptions of mathematics which had begun in the mid-seventeenth century. Focussing on one small part of this process, I aim to elucidate in detail how those uses and perceptions changed in England during approximately the period 1650-1750. This period encompasses major changes in the content of mathematics, including the invention of the calculus and the development and widespread adoption of symbolic algebra, as well as the appearance and reception of Newton's Principia and the initial development of the Royal Society's mathematising approach to natural phenomena. My previous work on the marginal mathematical practice of 'harmonics' in the late seventeenth century suggests that this period was important for the initial shift away from the concept of 'mixed mathematics' and towards that of 'applied mathematics'.

The classic accounts of the mathematical practitioners of Tudor, Stuart, and Hanoverian England by E.G.R. Taylor, and ongoing research on various specific mathematical practices, including my own doctoral work, provide the historiographical context for this project. My research will answer general questions such as the following:

Who used mathematics, and how did users of mathematics, distinct from 'mathematicians', identify themselves? How did they describe mathematics in the abstract, and how did they conceive it in actual use? How did the perceived relationship between mathematics and reality change over this period? And how did learned and popular writers arrive at the perception that certain mathematical practices had a reliable relationship with reality such that they could be 'applied' to a very wide variety of problems?

'Harmonics' illustrates some of the issues I expect to explore. The mathematical study of music was vexed in the seventeenth century by questions about the meaning of mathematical precision in relation to the aural experience of music and the fallibity of the ear, and the difficulty of producing satisfactory experiments or experimental instruments for the study of musical sound. Writers on mathematical music in the early eighteenth century could be familiar with both musical ratio theory based on ancient Greek models, 'harmonics', and the new mathematical science of acoustics pioneered by Newton's 1687 analysis of the sound wave. 'Harmonics' took harmony to be essential to each of the phenomena which embodied it as well as to the human mind, and was consequently primarily interested in pure reason. Acoustics, by contrast, found mathematical regularity in the accidental details of mechanical causes and human sense organs, and was interested primarily in experimental data as a source for theory-building and as a check on predictions. The case of music therefore displays one relationship between mixed science (harmonics) and applied mathematics (acoustics), and exemplifies some of the differences between the two.

I propose to approach this topic along two distinct routes. The first, already well under way, is the creation of a database of mathematical publications from the period in question. The use of modern bibliographic tools and databases promises to produce a catalogue perhaps twice as large as that of Taylor, with the possibility of efficiently extracting detailed statistics about chronological change in publication patterns. I have presented preliminary statistical results for the seventeenth-century material at a seminar on the history of mathematics in Oxford, and aim to complete this phase of the project during the first year of the project.

My second line of approach is through a series of case studies, looking in detail at particular mathematical practices. During the second phase of the project I aim to complete studies on the 'learned' practices of 'harmonics', optics, architecture, and perspective.

During the third phase of the project I intend to complete case studies on less learned uses of mathematics, including mensuration, dialling, and astronomy/astrology. Different detailed patterns will emerge, but I expect to find similarly revealing uncertainties about the meaning and proper use of mathematics. I will also examine those published texts in which authors make humorous or figurative use of mathematical terms, usually in order to make a polemical point about their authority or that of others. These, and all the texts I will consider, can shed much light on the perceived status of mathematics during this transitional time.

The outcome of this project will be a substantial scholarly monograph, which will make a significant contribution to our understanding of the emergence of the modern concept of 'applied mathematics' and the routes taken to it from the Renaissance 'mixed sciences'.